2000/03/20 by Stephen D. Miller, Miller, Stephen D., Tonghai Yang +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT
paper · pdf · doi:10.48550/arxiv.math/0003114
19 Pages
arxiv created 2000/03/20 · openalex publication_date 2000/03/20 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the early 1980s, Rohrlich began a study of canonical Hecke characters, which are closely related to the simplest examples of CM elliptic curves. He and Montgomery showed the non-vanishing of the central value when the L-function has an even functional equation, and we now show the non-vanishing of the central derivative when the functional equation is odd. Using the results of Gross-Zagier and Kolyvagin-Logachev, we can apply the non-vanishing to the ranks and Shafarevitch-Tate groups of the Q-curves "A(p)" studied by Gross in his thesis. In particular, their rank is determined by a congruence condition. http://www.math.yale.edu/users/steve/milleryang